I am aware of various numerical methods for solving definite integrals like Gauss Quadrature rule, Traezoidal rule, Simpson rules etc. But I could not find any such numerical methods in the literature for approximating indefinite integrals.
Question: Are there any approximation methods for solving indefinite integrals?
Sure... You can approximate the integral $F(x) = \int_a^x f(t) dt$ by any numerical quadrature of your liking. This way you can determine $F(x)$ for any particular $x$ using a quadrature on the interval $[a,x]$.